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Acknowledgment The authors wish to express their appreciation to Dr. M. S. Graboski for his helpful discussion and suggestions. Nomenclature B = second virial coefficient f = fugacity P = pressure P, = critical pressure P, = reduced pressure R = gas constant T = temperature T , = critical temperature T , = reduced temperature V = specific volume V , = critical volume Yi= composition of component i Z, = critical compressibility factor
w
549
= acentric factor
Literature Cited Canjar, L. N., Manning, F. S.,"Thermodynamic Propertiis and Reduced Corrections for Gases", Gulf Publishing Co., Houston, Texas, 1967. Chueh, P. L., Prausnitz, J. M., AIChE J . , 13, 1107 (1967a). Chueh, P. L.. Prausnitz, J. M., Ind. Eng. Chem. Fundam., 6, 492 (1967b). Lee, B. I., Kesler, M. G., AJChE J., 21, 510 (1975). Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibrium", p 148, Prentice-Hall, Inc., Englewood, Cliffs, N.J., 1969. Redlich, O., Kwong, J. N. S., Chem. Rev., 44, 233 (1949). Reid, R. C., Prausnitz, J. M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed. Chapter 6, McGjaw-Hill, New York, N.Y., 1977. Smith, J. M., Van Ness, H. C., Introduction to Chemical Engineering Thermodynamics", 3rd ed, Chapter 7, McGraw-Hill, New York, N.Y., 1975. Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Starling, K. E., "Fluid Thermodynamic Properties for Light Petroleum Systems", Gulf Publishing Co., Houston, Texas, 1973. Tsonopoulos, C., AIChEJ., 20, 263 (1974).
Received for review August 22, 1977 Accepted M a y 2, 1978
Selection of Measurements for Optimal Feedback Control D. J. Mellefont and R. W. H. Sargent" Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, London SW7 2BY, England
A stage in control system design which has received little attention in the literature is the selection of measurements to be used in feedback control. In the chemical industry there are often a number of alternative means of providing information for control and a large number of potential measurement locations. Measurements are costly and subject to random errors and must be chosen to satisfy hardware and budget constraints. In this paper, an implicit enumeration algorithm is presented for the selection of measurements to be used in optimal feedback control of a linear stochastic system. The algorithm is used to find the optimal location of measurements for control of a binary distillation column. Results show the importance of taking into account the effect of measurement information on control.
Introduction Feedback of measurement information is a necessary feature of control systems to ensure that control objectives are met and disturbances to the process can be compensated. Control system design therefore involves selecting measurements which are costly and subject to random errors. In many situations a number of alternative measurements can be considered. The compositions of process streams can be measured directly or inferred from related physical properties such as temperature, pH, or refractive index. To complicate the selection, multistage processes common in the chemical industry provide a large number of possible measurement locations and the choice of measurements must also satisfy hardware or budget constraints. The problem of selecting measurements for feedback control is combinatorial in nature. Previous attempts to solve this problem have resulted in heuristic design techniques. Weber and Brosilow (1972) use static deterministic models and present a design method to account for the influence of measurement accuracy on the estimation of state variables. They do not consider the effect that feedback of the measurement information has on system control. Kafarov et al. (1973) use an information theoretic approach to locate measurements but are only concerned that the region of uncertainty about the model is minimized. The application of measurement infor0019-7882/78/1117-0549$01.00/0
mation to feedback control must be taken into account when evaluating the measurements. In this paper, an implicit enumeration algorithm is developed for selection of measurements to be used in optimal feedback control of a linear stochastic system. The more general problem of allowing measurement subsets to change during the control interval is treated by Mellefont and Sargent (1977). Problem Formulation Consider a linear stochastic system dx = [AX + Buldt + dwl (1)
E(x(to)]= xO; covIx(to)l = Q o E(dw1) = 0; cov{dwJ = Vi dt with m possible measurements satisfying dz = CX dt + dw, (2) z(to) = 0; E{dw,J = 0; cov(dw,] = Vz d t where dwl and dw, are Wiener processes. The control objective is to minimize a quadratic performance index
J = E{ i o t f x B l x+ u B 2 u dt
+ xIt,)Pfx(tf)\
(3)
For a given set of measurements, the solution to this problem is well known (Kwakernaak and Sivan, 1972) with 0 1978
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optimal control u , given by u = -R 2 -1B'PR
(4)
where P is given by the control Riccati equation
-P = A'P
+ P A + R1 - PBR2-'B'P
P(tJ = P f (5) and 2 is the state estimate provided by the Kalman filter d2 = [ A i + Buldt
+ QC'V2-l [dz - CR dt]
(6)
where Q is the estimation error covariance and satisfies
Q = A Q + Q A ' + VI - QC'V2-'CQ
Q(t0) = Qo
(7)
The objective function reduces to
2 = xo'P(t,)xo + Tr[P(to)Qo+ J t f P V ,
+ PBR2-'B'PQ dtl
(8)
0
Equations 4-8 apply for whatever set of measurements have been selected and the optimal set of measurements must minimize (8) subject to measurement costs and constraints. Define a selection vector s such that si = 1 (if ith measurement used) s, = 0
(if not)
(9)
Then the cost of measurement can conveniently be represented by a quadratic function s'Rg. Off diagonal terms in R3 allow for any extra costs or discounts which are incurred when using any given combination of measurements. For example, use of multipoint measuring devices provides measurement facilities for extra measurements at little extra cost. The objective function for measurement selection can therefore be written
J = s B 3 s + Tr Lot>BR2-'B'PQ dt
(10)
Other terms in (8) can be neglected since they are independent of the choice of measurement set. It is interesting to note that the measurement selection problem is thus independent of the initial plant state xo. The use of measurement information in feedback is reflected in the term Tr.PBR2-'B'PQ which was shown by Mellefont (1977) to represent the additional control cost incurred by having to rely on any estimate of state rather than its exact value. For steady-state control problems integration in (10) is removed and P and Q are solutions to the appropriate algebraic Riccati equations. Implicit Enumeration The problem is to choose a set of measurements which minimizes (10) subject to ( 5 ) and (7). With m possible measurements, there are 2'" possible combinations and enumeration of all possible combinations is only practical for small problems. Since the integer variables si are binary, implicit enumeration techniques are particularly useful. The efficiency of these techniques depends largely on obtaining good upper and lower bounds on the objective (10) and on the method used for branching. Heuristic rules can be found by examination of problem structure to develop efficient implicit enumeration algorithms. A lower bound on the optimum can be found by assuming measurement costs are zero. For measurement design problems of interest, Q will be minimal with full measurement. Therefore a lower bound on objective (10) is given by
where K = PBR2-'B'P. A particular solution corresponding to full measurement can be easily calculated by adding measurement costs. If needed, an absolute upper bound can be found by calculating J with s, = 0 for all i and adding the cost of full measurement. The next stage is to select a branching variable s, to create two sub-problems corresponding to s, = 1 and s, = 0. Since the lower bound was calculated assuming full measurement, a branch can be made about that measurement which is most costly. However, in situations where equally costly measurements occur, such as multistage processes, this method is not a sufficient basis for selection. A measure of the instantaneous information provided by the measurement i can be found in the decrease in objective realized. G(Tr.KQ) Tr.KGQ = Tr.KQC,'V2,-1C,Q (12) Let p, be an average of (12) taken over the control interval; then the branching variable is chosen to be the measurement with the maximum value of A,s'R.~s- @, (13) where A, represents a difference s, = 1 to s, = 0, s, = 1 unless already set to zero (j# i), and cy is some positive scale factor. Having selected a measurement about which to branch, two sub-problems are created with s, = 1 and s, = 0, respectively. For the s, = 1problem, the particular solution remains unchanged and a new lower bound can be calculated by simply adding the appropriate measurement cost. This is a big advantage since (11)need not be recalculated. The s, = 1 sub-problem is put aside as least promising to be tested a t a later stage. A new lower-bound is calculated for the s, = 0 sub-problem. Unless a better estimate is available, the initial upper bound is taken as an estimate J' of the minimum objective function value. Whenever a feasible particular solution is obtained, it is compared with the current estimate 2 and if lower in magnitude provides a new estimate of the optimum. When the calculated lower bound is greater than Jo,the branch can be terminated. A new sub-problem must be selected and since the lower bounds have already been calculated, it is possible to search among the set of sub-problems and choose the problem with least lower bound. Constraints Combinatorial constraints such as an imposed maximum or minimum number of measurements or inadmissible combinations can be easily incorporated into the implicit enumeration algorithm. Initially, all integer variables s, are unspecified. As branches are made, variables are set to either zero or one and only those sub-problems which represent sub-sets of feasible combinations are considered. When it is no longer possible to form a feasible combination, the branch is terminated. The implicit enumeration algorithm is summarized in the Appendix. Measurement Selection for Control of a Binary Distillation Column Distillation is a multistage process common in the chemical industry. A large number of possible measurement locations exist but costs and hardware constraints prohibit taking measurements on each plate. The nine-plate binary distillation column example of Pollard and Sargent (1970) is considered here. It is assumed that the column will be operated about a steady state using optimal feedback control and the model equations were
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
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Table I. Measurement Locations for Optimal BDC Control (System: 9-Plate Binary Distillation Column with Feed on Plate 5. Measurement of Concentration on Each Plate, Cost = 0.1, Variance = 0.0004) mod- reboiler el plate 1 weightings control gain policy covariance
50 27.6 1 0.003
5 1.04 0 0.037
5 1.24 1 0.021
5 0.87 0 0.037
5 0.56 0 0.037
5 0.67 0 0.040
5 0.80 1 0.017
condenser 9 5 50 0.81 0.19 0 0 0.04 7 0.005
weightings control gain policy covariance
50 27.6 1 0.003
5 1.04 0 0.037
5 1.25 1 0.022
5 0.87 0 0.039
5 0.56
5 0.63 0 0.048
5 0.60 0 0.045
5 0.32 0 0.041
50 2.00 1 0.001
weightings control gain policy covariance
50 32.8 1 0.003
0
0 0.32 1 0.022
0 0.20 0 0.040
0
0.08 0 0.041
0 0.025 0 0.054
0 0.043 0 0.055
0 0.15 0 0.054
50 0.092 0 0.006
weightings control gain policy covariance
50 32.8 1 0.003
0 0.38 0 0.037
0
0 0.20 0 0.042
0 0.08 0 0.044
0 0.015 0 0.068
0
0.32 1 0.022
0 0.005
0.080
0 0.079
50 0.96 0 0.031
weightings control gain
50 19.8
5 4.82
5 1.18
5 0.35
5 0.35
5 0.67
5 0.82
5 0.82
50 0.19
weightings control gain
50 20.7
0 3.80
0 0.59
0
0 0.015
0 0.015
0 0.042
0 0.15
50 0.090
2
0.38 0
0.037
3
4
5
0 0.039
0.08
model
reboiler column condenser 32.0 625.0 64.0 625.0 625.0 64.0 64.0 64.0 32.0 control gain = diagonal elements of PBR,-’B’P matrix at steady state. covariance = diagonal elements of final time covariance Q( t f ) .
6
7
0.005 0
8
holdups
1 2 3
linearized about this state. A short control period of 0.5 h was specified which effectively gives a high weighting to the initial period where maximum measurement effort is needed to reduce a high covariance, taken to be 0.11 (I is the unit matrix). The implicit enumeration algorithm was used to solve a series of problems and the optimal policies are presented in Table I. On average less than 100 out of a possible 512 combinations were enumerated. Concentrations at the reboiler and condenser were given the highest weighting in matrix R1 of eq 3 since they reflect product quality. To ensure that concentrations on intermediate plates do not vary excessively, some lesser weighting is given to these concentrations for two of the problems. The first example shows an optimal location of three measurements with an emphasis on the bottom end of the column. A relatively high reboiler holdup was specified which results in a high control gain, R2-lB’P,with respect to bottoms concentration. Deviations from set point respond slowly to control corrections and therefore an accurate estimate is beneficial at this point. On the other hand, condenser holdup is low and it appears that better control results from measuring on the seventh plate. Increasing condenser holdup in the second example moves this measurement to the top of the column. Repeating these examples with zero weighting on intermediate plates removes the measurement from the top end of the column. The unusual feature of these examples is that optimum locations remain unchanged (and at the bottom end of the column) even when condenser holdup is increased. Variances of top and bottom product concentrations are different by an order of magnitude. To explain this result, the interpretation of the objective (10) given previously can be used. The problem is seen as one of balancing measurement cost against the additional control cost incurred by having to rely on an estimate of state. The appropriate weighting matrix PBR2-’B’P was approximately constant over a large segment of the control
interval, and diagonal elements of this matrix at steady state are included in Table I. This shows the relative importance of estimation error on the objective function and serves to justify the optimal policies obtained. Even when the reboiler holdup is low (model 3), control costs are dominated by deviations in concentration at the bottom end of the column. Shunta and Luyben (1971) have studied the effect of measurement location on reflux control (conventional PID) of a binary distillation column. They found that trays a t the top of the column have a slower response to changes in reflux than do lower ones which exhibit a correspondingly greater change in composition. Conversely, it is much more difficult to detect a deviation from set point at the top of the column than lower down. Accurate estimation of concentration on lower trays enables correct control action to be taken before significant deviations in product quality occur. Shunta and Luyben also found that trays near the feed tray approach equilibrium faster than those at the ends of the column when a step change is made in the reflux. Although using optimal state feedback control as opposed to conventional single loop control, the examples of their Table 2 show a minimum control cost weighting on the feed plate (no. 5 ) . Results therefore show that accurate estimates are needed of those states which have the highest control gains. Conclusions These examples show the effectiveness of the proposed implicit enumeration algorithm for the selection of measurement locations in multistage processes. Similarly, the algorithm can be used to select between different methods of measuring a given state variable. The ultimate use of measurements for feedback control is reflected in the solutions obtained which are considerably different from what would be expected for pure estimation. Appendix Algorithm for Optimal Measurement Selection by Implicit Enumeration. 1. Generate the first element
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in the list of sub-problems with si= 1 for all i and calculate J from (11). If this particular solution is feasible, add on costs to give d; otherwise, use the absolute upper bound. Set Jo = d. 2. Take the sub-problem with least lower bound J from the list. If the list is empty then JO is optimal. If JO equals the absolute upper bound then no feasible solution exists. 3. If J > JO go to step 2. 4. If J < Jo set JO = d and record particular solution. 5. If no more feasible combinations are possible, go to step 2. 6. Select the unspecified measurement i with largest cost (13) that gives a feasible combination when eliminated. Add a sub-problem to the list with si = 1 (provided that the specified combination is feasible) and with lower bound J + sX3s. d remains unaltered.
7. Set si = 0 and calculate J from (11). If the particular solution is feasible add on measurement cost to give J, otherwise d is set to the absolute upper bound. Go to step 3. Literature Cited Kafarov,V. V., et ai., Theor. Found. Chem. Eng. (USSR), 7, 224-229 (1973). Kwakernaak, H., Sivan, R., "Linear Optimal Control Systems", Wiley, New York, N.Y., 1972. Mellefont, D. J., Ph.D. Thesis, University of London, 1977. Meilefont, D. J., Sargent, R. W. H., "Proceedings of the 8th IFIP Conference on Optimization Techniques, Wuzburg, 1977",J. Stoer, Ed., Vd. I, pp 166176, Springer-Verlag, Berlin, 1978. Pollard, G. P., Sargent, R. W. H.,Automatica, 8, 59-76 (1970). Shunta, J. P., Luyben, W. L., AIChEJ., 17 ( l ) , 92-97 (1971). Weber, R., Brosilow, C., AIChE J., 18 (3), 614-623 (1972).
Received for review October 25, 1977 Accepted April 24, 1978
Application of the UNIQUAC Equation to Calculation of Multicomponent Phase Equilibria. 1. Vapor-Liquid Equilibria T. F. Anderson and J. M. Prausnitr" Department of Chemical Engineering, University of California, Berkeley, California 94 720
Pure-component UNIQUAC parameters have been calculated for 90 fluids and binary parameters have been obtained for 142 binary systems. All binary parameters were found from data reduction using the principle of maximum likelihood. The UNIQUAC equation is modified slightly to yield better results for those binary systems where an alcohol is one of the components. As in all equations for excess Gibbs energy, UNIQUAC parameters are not unique and they are at least weakly temperature dependent. The effects of parameter uncertainty are fortunately not large for typical vapor-liquid equilibria. Illustrative examples are given for a variety of strongly nonideal binary and ternary mixtures.
Because of its importance in design of separation operations, chemical engineers have given much attention to the thermodynamics of phase equilibrium in fluid mixtures. While numerous semiempirical equations have been proposed for calculating activity coefficients in mixtures of nonelectrolytes, a truly satisfactory equation has not as yet been established. However, a particularly useful equation, applicable to a wide variety of liquid mixtures, was given by Abrams and Prausnitz (1975); this equation, called UNIQUAC, uses only two adjustable parameters per binary in addition to pure-component parameters reflecting the sizes and outer surface areas of the molecules. In this work we consider in detail how the UNIQUAC equation can be used to represent binary and multicomponent vapor-liquid equilibria in typical mixtures of nonelectrolytes at low or moderate pressure. Toward that end we briefly review a recently developed method for reduction of experimental data to obtain UNIQUAC parameters; next, we propose an empirical modification in UNIQUAC which significantly improves the ability of UNIQUAC to represent the properties of mixtures containing alcohols, and finally, we present a variety of examples to illustrate how UNIQUAC may be used to correlate binary and predict multicomponent vapor-liquid equilibria. In the second article, immediately following 0019-7882/78/1117-0552$01.00l0
this one, we consider multicomponent liquid-liquid equilibria. Fundamental Thermodynamic Relations
When a liquid phase is in equilibrium with a vapor phase, the compositions of the two phases are related by a set of equations, one for each component i
@yip= yixifio
(1)
where x is the liquid-phase mole fraction, y is the vapor-phase mole fraction, @ is the vapor-phase fugacity coefficient, y is the liquid-phase activity coefficient, P is the total pressure, and f" is the standard-state fugacity. The vapor-phase fugacity coefficients are calculated from an equation of state. At low or moderate pressures we use an equation of state which considers only two-body interactions; for mildly interacting components, this is the virial equation truncated after the second term and for strongly interacting systems (e.g., those containing a carboxylic acid), this is an equation based on a (chemical) dimerization hypothesis. Our method is that given by Hayden and O'Connell (1975) as briefly discussed in Appendix A. Liquid-phase activity coefficients are related to the molar excess Gibbs energy gE through
0 1978 American Chemical Society
